Transcript System of equations

9.1
Solving Systems of Linear Equations
Graphically
1. Determine whether an ordered pair is a solution for a
system of equations.
2. Solve a system of linear equations graphically.
3. Classify systems of linear equations in two unknowns.
System of equations: A group of two or more
equations.
 x y 5

3x  4 y  8
(Equation 1)
(Equation 2)
Solution for a system of equations: An ordered pair
that makes all equations in the system true.
To Check a Solution to a System of Equations
1. Replace each variable in each equation with its
corresponding value.
2. Verify that each equation is true.
Determine whether the ordered pair (3, 4) is a solution
to the system of equations.
x  y  7

 y  3x  2
(Equation 1)
(Equation 2)
x+y=7
3+4=7
7=7
True
y = 3x – 2
4 = 3(3) – 2
4=7
False
Because (3, 4) does not satisfy both equations, it is not
a solution to the system of equations.
Determine whether the ordered pair (3, 2) is a
solution to the system of equations.
x  y  7

 y  3x  2
(Equation 1)
(Equation 2)
x+y=7
3 + 2 = 7
1 = 7
False
y = 3x – 2
2 = 3(3) – 2
2 = 11
False
Because (3, 2) does not satisfy both equations, it is not a
solution for the system.
Which set of points is a solution to the
system?
2 x  4 y  2

3x  2 y  1
a) (–1, 1)
b) (–1, –1)
c) (0, 2)
d) (–3, 7)
9.1
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Slide 5- 6
Which set of points is a solution to the
system?
2 x  4 y  2

3x  2 y  1
a) (–1, 1)
b) (–1, –1)
c) (0, 2)
d) (–3, 7)
9.1
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Slide 5- 7
A system of two linear equations in two variables can
have one solution, no solution, or an infinite number of
solutions.
x  y  5

 y  2x  4
The graphs intersect
at a single point.
There is one
solution.
 y  3x  1

 y  3x  2
The equations have
the same slope, the
graphs are parallel.
There is no solution.
x  2 y  4

2 x  4 y  8
The graphs are
identical. There are
an infinite number
of solutions.
(Not the same as all real numbers.)
Solving Systems of Equations Graphically
1. Graph each equation.
a. If the lines intersect at a single point, then the
coordinates of that point form the solution.
b. If the lines are parallel, there is no solution.
c. If the lines are identical, there are an infinite
number of solutions. They are the coordinates
of all the points on that line.
2. Check your solution.
Solve the system of equations graphically.
y  2  x

2 x  4 y  12
(Equation 1)
(Equation 2)
Graph each equation:
(2, 4)
2x + 4y = 12
y=2–x
y = -x + 2
(0,2)
m = -1
2x + 4y =12
(0,3) (6,0) m = -½
Intersection:
The solution is (-2,4).
y=2–x
Solve the system of equations graphically.
3

y  x 3
4


3x  4 y  4
(Equation 1)
(Equation 2)
 4 y  3x  4
3
y  x 1
4
The slopes are the
same, so the lines are
parallel. The system
has no solution
Solve the system of equations graphically.
4 x  2 y  2

 y  2x 1
(Equation 1)
(Equation 2)
 2y  4 x  2
y  2x  1
The equations are identical. All
ordered pairs along the line are
solutions.
Solve the system of equations graphically.
y  x  4

x  2y  2
a) (2, -3)
b) (2, -2)
c) No Solution
d) Infinite number of solutions
9.1
Copyright © 2011 Pearson Education, Inc.
Slide 5- 13
Solve the system of equations graphically.
y  x  4

x  2y  2
a) (2, -3)
b) (2, -2)
c) No Solution
d) Infinite number of solutions
9.1
Copyright © 2011 Pearson Education, Inc.
Slide 5- 14
Consistent system of equations: A system of
equations that has at least one solution.
Inconsistent system of equations: A system of
equations that has no solution.
Classifying Systems of Equations
Consistent system with
independent equations:
Consistent system with
dependent equations:
The system has a single
solution at the point of
intersection.
The system has an infinite
number of solutions.
The graphs are different.
They have different slopes.
Inconsistent system:
The system has no
solution.
The graphs are identical.
The graphs are parallel
lines.
They have the same slope
and same y-intercept.
They have the same slope,
but different y-intercepts.
Copyright © 2011 Pearson Education, Inc.
Classify this system of equations.
a) Consistent with
independent equations
b) Consistent with
dependent equations
c) Inconsistent
9.1
y  2  x

2x  4 y  12
(2, 4)
2x + 4y = 12
y=2–x
Classify this system of equations.
a) Consistent with
independent equations
b) Consistent with
dependent equations
c) Inconsistent
9.1
y  2  x

2x  4 y  12
(2, 4)
2x + 4y = 12
y=2–x
Classify this system of equations.
a) Consistent with
independent equations
b) Consistent with
dependent equations
c) Inconsistent
9.1
3

y  x  3
4

3x  4 y  4
Classify this system of equations.
a) Consistent with
independent equations
b) Consistent with
dependent equations
c) Inconsistent
9.1
3

y  x  3
4

3x  4 y  4
Classify this system of equations.
a) Consistent with
independent equations
b) Consistent with
dependent equations
c) Inconsistent
9.1
4 x  2y  2

y  2 x  1
Classify this system of equations.
a) Consistent with
independent equations
b) Consistent with
dependent equations
c) Inconsistent
9.1
4 x  2y  2

y  2 x  1